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Replacing the concept of distance by another, also involving
*imaginary' elements, Cayley provided the means for unifying
Euclidean geometry and the common non-Euclidean geometries
into one comprehensive theory. Without the use of some
algebra it is not feasible to give an intelligible account of how
this may be done; it is sufficient for our purpose to have noted
Cayley's main advance of uniting protective and metrical
geometry with its cognate unification of the other geometries
just mentioned.
The matter of n -dimensional geometry when Cayley first put
it out was much more mysterious than it seems to us to-day,
accustomed as we are to the special case of four dimensions
(space-time) in relativity. It is still sometimes said that a fonr-
dimensional geometry is inconceivable to human beings. This
is a superstition which was exploded long ago by Pliicker; it is
easy to put four-dimensional figures on a flat sheet of paper,
and so far as geometry is concerned the whole of a four-dimen-
sional 'space' can be easily imagined. Consider first a rather
unconventional three-dimensional space: all the circles that
may be drawn in a plane. This 'all' is a three-dimensional
'space' for the simple reason that it takes precisely three
numbers, or three co-ordinates, to individualize any one of the
swarm of circles, namely fero to fix the position of the centre
with reference to any arbitrarily given pair of axes, and one to
give the length of the radius.
If the reader now wishes to visualise a four-dimensional space
he may think of straight lines, instead of points, as the element
out of which our common 'solid* space is built. Instead of our
familiar solid space looking like an agglomeration of infinitely
fine birdshot it now resembles a cosmic haystack of infinitely
thin, infinitely long straight straws. That it is indeed four-
dimensional in straight lines can be seen easily if we convince
ourselves (as we may do) that precisely four numbers are neces-
sary and sufficient to individualize a particular straw in our
haystack. The 'dimensionality' of a fcspaee* can be anything we
choose to make it, provided we suitably select the elements
(points, circles, lines, etc.) out of which we build it. Of course
if we take points as the elements out of which our space is to be